Mixed Tenney-Weil norms and the dual to Cangwu badness

These are incomplete thoughts, but I'm putting them here for reference as
I'll ultimately come back to these.

I did some work on mixed Tenney-Weil norms a long time ago, which I think
are documented in the archives here under the name "Tenney-Farey" norm.

One thing I'd worked out is a very, very general Tenney-Weil Lp norm, which
had two free parameters. That norm is defined, for some full-limit
*weighted* monzo m = |e_2 e_3 e_5 ...>, as:

(|e_2|^p + |e_3|^p + |e_5|^p + ... + k*|e_2+e_3+e_5...|^p)^(1/p)

Note that the last term, which is the sum of the weighted coordinates, is
the span of the monzo.

There are two free parameters here, denoted k and p. The k parameter tells
you how much you want the span to count, and p is the exponent of the
p-norm of the result.

To recap some of this, you can write this as a norm on "augmented monzos,"
where there's an additional coordinate at the end corresponding to the span
of the monzo. This is a useful perspective because then you get a pure Lp
norm on this larger space of "augmented monzos," and the embedding from
normal monzos to embedded ones is trivial. You can also use this principle
to work out the restricted norm for any subgroup.

This also induces a dual norm on the dual space of vals, and likewise can
induce norms on the exterior algebra of both of these spaces. However, I
recall that unless p=2 the extension to the exterior algebra may not be
unique. (Need to rehash notes on this, but it's a tangent)

Either way, this norm is interesting because:

1) For k=0, you get the Tp norm
2) For k=1, you get the Wp norm
3) For k=0, p=1, you get the Tenney norm
4) For k=0, p=2, you get the TE norm
5) For k=1, p=1, you get the Weil norm
6) For k=1, p=2, you get the WE norm (Weil-Euclidean), which is interesting
and has not yet been fully explored

Here's another nice thing that I also think is true:

7) For p=2, and letting k vary, you get the dual norm to Cangwu badness,
perhaps scaled by a constant factor

Here's why. Cangwu badness elongates the TE unit sphere in the direction of
the JIP. This thing, in contrast, penalizes intervals according to span.
The axis corresponding to "interval span" is basically the monzo version of
the JIP, meaning the monzo with the same coordinates as the JIP. So you end
up compressing the unit sphere around the same axis that the JIP elongates
it.

Of, if you like, you can think of it as elongating the unit sphere in every
direction that's orthogonal to this monzo-JIP, the set of which corresponds
to what Paul has called a "pitch contour" - specifically the one going
through 1/1.

===

This leads to a few interesting thoughts, right off the bat:

1) There should be some special value of Ek for Cangwu badness that
corresponds exactly to the Weil-Euclidean norm.

2) If you instead fix p=1 and let k vary, you get a new family of norms
which contains Tenney and Weil instead of TE and WE.

3) The dual norms induced by this family on the dual space of vals will be
some sort "bastard" child of TOP and Cangwu badness. They're the Linf-ified
version of Cangwu badness. They minimize the max (Tenney+k*span)-weighted
error over all monzos, with k=1 corresponding to Weil. The more you count
the span, the more you care about error vs complexity.

4) One of these will correspond to the dual of the ordinary Weil norm,
which is an octave-inequivalent version of the thing Gene has called
"generator complexity."

5) There was one alternative way to formalize the generalization of the
Weil norm to arbitrary Lp, and I should check that those results sync up
with these, or that these are unique in some sense.

Something to come back to later...

Mike